\select@language {ngerman}
\contentsline {chapter}{Abbildungsverzeichnis}{II}{chapter*.5}
\contentsline {chapter}{\numberline {1}Einleitung}{1}{chapter.6}
\contentsline {section}{\numberline {1.1}Ein Historischer R"uckblick}{1}{section.7}
\contentsline {section}{\numberline {1.2}Voraussetzungen}{3}{section.20}
\contentsline {chapter}{\numberline {2}Halbebenenmodell der hyperbolischen Geometrie}{5}{chapter.21}
\contentsline {section}{\numberline {2.1}Die $\mathcal {H}$-L"ange einer Strecke}{6}{section.24}
\contentsline {section}{\numberline {2.2}Entwicklung der Geod"ate}{7}{section.29}
\contentsline {section}{\numberline {2.3}Die $\mathcal {H}$-Gerade}{10}{section.38}
\contentsline {chapter}{\numberline {3}Die Inversion am Kreis}{14}{chapter.53}
\contentsline {section}{\numberline {3.1}orthogonale Kreise und Winkeltreue}{15}{section.59}
\contentsline {chapter}{\numberline {4}M"obiustransformationen}{20}{chapter.70}
\contentsline {section}{\numberline {4.1}Invarianzen und erhaltene Strukturen}{22}{section.75}
\contentsline {subsection}{\numberline {4.1.1}Doppelverh"altnis}{23}{subsection.78}
\contentsline {chapter}{\numberline {5}Bewegungen}{27}{chapter.86}
\contentsline {section}{\numberline {5.1}H-Bewegungen}{29}{section.90}
\contentsline {section}{\numberline {5.2}Bewegungs-Typen}{30}{section.92}
\contentsline {section}{\numberline {5.3}dreifach Spiegelungen}{34}{section.109}
\contentsline {chapter}{\numberline {6}Abschluss}{37}{chapter.113}
\contentsline {chapter}{\numberline {7}Notationen}{38}{chapter.114}
\contentsline {chapter}{Literaturverzeichnis}{39}{chapter*.115}
\contentsline {chapter}{Eidesstattliche Erkl\"arung}{41}{chapter*.116}
\contentsline {chapter}{\numberline {A}Anhang}{i}{appendix.117}
